Homework help 3 – reducing fractions method 1

(The third in a semi-regular series. A math boot camp to help you help your kiddo, if you will.)

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Because I know that’s what all the non-geeky just did at the word “fractions”.  One more time – get it out of your system:

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA!!!!!!!!!!!!!!!!!!!!!!

Ok.  Let’s start (like always) with some of the vocab:

  • Denominator: (my students have nominated this for my most frequently misspelled board word).  This is the bottom of a fraction.  In a math situation the denominator tells you how many pieces you have cut a unit into.  Because I have an ongoing love-affair with pizza, most of our examples will have to do with this delicious food of the gods.  The denominator tells you how many pieces you cut a pizza pie into.
  • Numerator: This is the top of a fraction.  In a math situation, the numerator tells you how many pieces you have.  In a pizza example, the numerator would be how many pieces of pizza you ate.

So if I ordered a pizza on the way home from teaching, but by the time I got home the hubs had eaten some of it, we could represent this with a fraction.  Say he ate 7/8… This means that the no-good pizza stealing hungry fellow ate 7 of the 8 pieces that the pizza was cut into.

If I had ordered 2 pies, each cut into 8 pieces, he would have eaten 13/8.  Notice that my denominator still says how many pieces EACH pizza was cut into.  The fact that the top is bigger than the bottom means that he ate one whole pizza plus some more of the next one.  Pig.  (Any fraction with a top the same size or bigger than the bottom is called an improper fraction.  9/8, 8/8, 17/8 : all improper.  I’ve been known to call them Dolly Parton fractions, for obvious reasons.  LOVE you, Dolly! :-*)

Equivalent fractions are fractions that tell you the same amount, but cut into different kinds of pieces.  Picture this:

Now, who ordered the orange and grape pizzas? Seriously! That’s the LAST time I let the toddler pick the family dinner.

If I eat 2 slices of orange pizza and you ate 4 slices of grape pizza, notice we ate the same amount of pizza.  Using our vocab from above, notice that I just ate 2/4 of a pizza and you just ate 4/8 of a pizza.  But we ate the same amount – my two big pieces are the same as your 4 little pieces.  (Now, where is that Pepto… Orange and grape pizzas my ass.)

Because we’re not going to want to draw pizzas for every math problem, there’s a handy arithmetic rule: two fractions are equivalent (or talk about the same amount of pizza) when the numerator and denominator of the first can be multiplied or divided by the same number to get to the second.  2/4 is equivalent to 4/8 because I multiplied the top and bottom of 2/4 both by 2.

Sometimes these homework problems will look like this: give an equivalent fraction to 2/3 with a denominator of 12.  What the homework gods are asking you is this: 2/3 = ?/12.  What did we multiply 3 by to get to 12? 4!  So we have to multiply the top, 2, by 4 also: 2 x 4 = 8.  So 2/3 = 8/12.

Let’s do another one, with division: give an equivalent fraction to 8/20 with a denominator of 5.  Again, what the homework gods are asking you is 12/30 = ?/5.  What did we divide 30 by to get to 5? 6!  So we have to divide the top, 12, by 6 also: 12 ÷ 6 = 2.  So 12/30 = 2/5.

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Reducing fractions to lowest terms is just finding an equivalent fraction – you want to divide the top and bottom until there’s nothing else that goes into both of them.  The nice thing is, as long as you do your times tables right, it doesn’t matter if you do this in 1 step or 17 – you’ll get the same answer at the end.

Lets take 15/30.  I would look at that and say Ooo! Ooo!  (To which the toddler would reply by running around doing his monkey impression.  Smart-ass.) 5 goes into both 15 and 30.  So I divide the top and bottom by 5, 15 ÷ 5 = 3 and 30 ÷ 5 = 6, so now I have 3/6.  What goes into both 3 and 6?  3! 3 ÷ 3 = 1 and 6 ÷ 3 = 2, so now I have 1/2.  Cutting a pizza into 30 pieces and eating 15 of them is the same as cutting it into 2 pieces and eating 1 of those.  Do 1 and 2 have anything else in common?  Nope!  Then the reduced form of 15/30 is 1/2.  Taa daa!!

Now, I could have done that faster by noticing that 15 goes into both 15 and 30, and in one fell swoop: 15 ÷ 15 = 1 and 30 ÷ 15 = 2, so now I have 1/2.  To be honest, I don’t stress too much about finding the biggest number to divide by.  I do these problems faster by dividing by the FIRST number that pops into my head that works, and dividing again if I have to.

Lets try a bigger one: 84/144.  Gah – that 84 has me stumped.  But I notice they’re both even, so I start by dividing top and bottom by 2: 84 ÷ 2 = 42, and 144 ÷ 2 = 72; so now we have 84/144 = 42/72.  Still even, so I divide top and bottom by 2 again: 42 ÷ 2 = 21, 72 ÷ 2 = 36; so now we have 84/144 = 21/36.  2 doesn’t work anymore because 21 isn’t even… but 3 goes into both 21 and 36!  21 ÷ 3 = 7 and 36 ÷ 3 = 12.  This gives us 84/144 = 7/12.  Does anything go into both 7 and 12?  Nope!  All done!  So the reduced form of 84/144 is 7/12.

(If you’re stuck on the “how do I know what goes into THAT” part – glance over at my homework help 1… it might help you out!)

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See!  You made it all the way through reducing fractions without spontaneous combustion!!  Next time we’ll talk about an easy way to reduce HUGE fractions, through factoring and cancelling 🙂 🙂 🙂

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